ch. 9110 Time Cost Time Cost r Immediate Predecessors Activity $3,000 7,000 3 $4

Question

ch. 9110 Time Cost Time Cost r Immediate Predecessors Activity $3,000 7,000 3 $4,000 2 4,600 5 $15,000 5 $15,000 5 2 $3,000 1 s 3,500 A,B,C C,D 2 $1,500 1 S 2,400 4 $4,0001 7,000 5 $7,000 4 $12,000 7 $8,000 5 $20,000 D, F Write the objective function and constraints to crash the above project to 14 units of time. Show work for any calculations and show work using algebra to change constraints, as needed, so all variables are on one side of each constraint. Do not solve. Do not identify the variables

Explanation / Answer

Determine allowable crashing limit for each activity and Crashing cost per period for each activity.

Activity

Predecessors

Normal Time (days)

Crash time (days)

Normal Cost

Crash cost

Allowable Crashing Limit

Crashing Cost per period

A

Start

5

1

3000

7000

4

1000

B

Start

3

2

4000

4600

1

600

C

Start

5

5

15000

15000

0

0

D

A, B, C

2

1

3000

3500

1

500

E

C, D

2

1

1500

2400

1

900

F

A

4

1

4000

7000

3

1000

G

D, F

5

4

7000

12000

1

5000

H

E

7

5

8000

20000

2

6000

LPP formulation

Let,

Xi = Finish time for activity i

Yi = amount of time activity i is crashed

Every activity should satisfy the relationship between finish time, earliest start time, and activity time as follows:

Finish time >= Earliest time + Activity Time = Earliest time + (Normal time – Crashed amount)

Activity

Comment

Equation

A

Starting activity

XA >= 0 + (5 – YA)

XA + YA >= 5

B

Starting activity

XB >= 0 + (3 – YB)

XB + YB >= 3

C

Starting activity

XC >= 0 + (5 – YC)

XC + YC >= 5

D

A is predecs.

XD >= XA + (2 – YD)

XD - XA + YD >= 2

D

B is predecs.

XD >= XB + (2 – YD)

XD – XB + YD >= 2

D

C is predecs.

XD >= XC + (2 – YD)

XD – XC + YD >= 2

E

C is predecs.

XE >= XC + (2 – YE)

XE – XC + YE >= 2

E

D is predecs.

XE >= XD + (2 – YE)

XE – XD + YE >= 2

F

A is predecs.

XF >= XA + (4 – YF)

XF – XA + YF >= 4

G

D is predecs.

XG >= XD + (5 – YG)

XG – XD + YG >= 5

G

F is predecs.

XG >= XF + (5 – YG)

XG – XF + YG >= 5

H

E is predecs.

XH >= XE + (7 – YH)

XH – XE + YH >= 7

Project has to be completed within 14 UNITS (G and H are finishing activities)

XG <= 14

XH <= 14

Maximum allowable crashing time for each activity

YA <= 4,

YB <= 1,

YC <= 0,

YD <= 1,

YE <= 1,

YF <= 3,

YG <= 1,

YH <= 2,

Non-negative constraint

All Xi and Yi >= 0

Activity

Predecessors

Normal Time (days)

Crash time (days)

Normal Cost

Crash cost

Allowable Crashing Limit

Crashing Cost per period

A

Start

5

1

3000

7000

4

1000

B

Start

3

2

4000

4600

1

600

C

Start

5

5

15000

15000

0

0

D

A, B, C

2

1

3000

3500

1

500

E

C, D

2

1

1500

2400

1

900

F

A

4

1

4000

7000

3

1000

G

D, F

5

4

7000

12000

1

5000

H

E

7

5

8000

20000

2

6000

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